Dirichlet series
| L(s) = 1 | + 0.621·2-s − 0.129·3-s + 0.523·4-s − 0.242·5-s − 0.0802·6-s + 0.733·7-s + 1.03·8-s + 1.05·9-s − 0.150·10-s + 0.0575·11-s − 0.0676·12-s + 1.22·13-s + 0.455·14-s + 0.0313·15-s + 0.0998·16-s − 1.19·17-s + 0.656·18-s + 0.534·19-s − 0.127·20-s − 0.0946·21-s + 0.0357·22-s + 0.0267·23-s − 0.133·24-s − 0.0738·25-s + 0.764·26-s − 0.399·27-s + 0.383·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+18.3i) \, \Gamma_{\R}(s+7.21i) \, \Gamma_{\R}(s-18.3i) \, \Gamma_{\R}(s-7.21i) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(11.2620\) |
| Root analytic conductor: | \(1.83191\) |
| Rational: | no |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((4,\ 1,\ (18.39113113456i, 7.210457668i, -18.39113113456i, -7.210457668i:\ ),\ 1)\) |
Euler product
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.9227334, −23.8406631, −22.3705951, −20.7096194, −15.7015578, −13.4238775, −11.1509662, −4.3065503, −1.5680358, 1.5680358, 4.3065503, 11.1509662, 13.4238775, 15.7015578, 20.7096194, 22.3705951, 23.8406631, 24.9227334